Optimal linear combination of order statistics filters and their relationship to the Delta-operator

Linear Combination of Order Statistics (LOS) filters are a special case of the Choquet integral filters. LOS are a class of nonlinear filters parameterized by a set of n weights. Different values of the weights lead to different filters. Examples include the median and other order statistic filters, local averaging filters, and trimmed average filters. Differences of LOS filters have been used in the past as target detection filters by nonlinearly comparing a small, targets size region with the surrounding region. The delta operator, proposed by Gelenbe et. al. for land mine detection, can be represented as a special case of a difference of LOS operators. Weights of LOS operators can be determined by solving an optimization problem, represented as a quadratic program. In this paper, experiments are conducted in determining optimal differences of LOS operators using the DARPA backgrounds data. The results are that the delta-operator is the solution of the optimization problem for this data set.